chap8 - 8. Discrete Multivariate Distributions 8.1 Basic...

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8. Discrete Multivariate Distributions 8.1 Basic Terminology and Techniques Many problems involve more than a single random variable. When there are multiple random variables associated with an experiment or process we usually denote them as X,Y,. .. or as X 1 ,X 2 ,... .Fo r example, your final mark in a course might involve X 1 =your assignment mark, X 2 =your midterm test mark, and X 3 =your exam mark. We need to extend the ideas introduced for single variables to deal with multivariate problems. Here we only consider discrete multivariate problems, though continuous multivariate variables are also common in daily life (e.g. consider a person’s height X and weight Y, or X 1 = the return from Stock 1, X 2 = return from stock 2). To introduce the ideas in a simple setting, we’ll first consider an example in which there are only a few possible values of the variables. Later we’ll apply these concepts to more complex examples. The ideas themselves are simple even though some applications can involve fairly messy algebra. Joint Probability Functions: First, suppose there are two r.v.’s X and Y , and define the function f ( x, y )= P ( X = x and Y = y ) = P ( X = x, Y = y ) . We call f ( x, y ) the joint probability function of ( X,Y ) . In general, f ( x 1 ,x 2 , ··· n P ( X 1 = x 1 and X 2 = x 2 and ... and X n = x n ) if there are n random variables X 1 ,...,X n . The properties of a joint probability function are similar to those for a single variable; for two r.v.’s we have f ( x, y ) 0 for all ( x, y ) and X all(x , y) f ( x, y )=1 . 115
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116 Example: Consider the following numerical example, where we show f ( x, y ) in a table. x f ( x, y ) 012 1 .1 .2 .3 y 2 .2 .1 .1 for example f (0 , 2) = P ( X =0 and Y =2 )=0 . 2 . We can check that f ( x, y ) is a proper joint probability function since f ( x, y ) 0 for all 6 combinations of ( x, y ) and the sum of these 6 probabilities is 1. When there are only a few values for X and Y it is often easier to tabulate f ( x, y ) than to find a formula for it. We’ll use this example below to illustrate other definitions for multivariate distributions, but firstwegiveashortexamplewhereweneedto find f ( x, y ) . Example: Suppose a fair coin is tossed 3 times. Define the r.v.’s X = number of Heads and Y =1(0) if H ( T ) occurs on the first toss. Find the joint probability function for ( X,Y ) . Solution: First we should note the range for ( ) , which is the set of possible values ( x, y ) which can occur. Clearly X canbe0 ,1 ,2 ,or3and Y can be 0 or 1, but we’ll see that not all 8 combinations ( x, y ) are possible. We can find f ( x, y )= P ( X = x, Y = y ) by just writing down the sample space S = { HHH,HHT,HTH,THH,HTT,THT,TTH,TTT } that we have used before for this process. Then simple counting gives f ( x, y ) as shown in the following table: x f ( x, y ) 0123 0 1 8 2 8 1 8 0 y 1 0 1 8 2 8 1 8 For example, ( )=(0 , 0) if and only if the outcome is TTT ;( )=(1 , 0) iff the outcome is either THT or TTH .
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chap8 - 8. Discrete Multivariate Distributions 8.1 Basic...

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